## Group automorphisms with few and with many periodic points.(English)Zbl 1052.37017

Let $$T:X\rightarrow X$$ be a compact group automorphism, $$F_n (T)=| \{ x\in X\mid T^n (x)=x\} |$$ be the number of points of period $$n$$. The main result of the paper is the following: for all $$C\in [0,\infty ]$$ there is a compact group automorphism $$T$$ with $$F_n (T)<\infty$$ for all $$n$$, and with $$\frac{1}{n}\log F_n (T)\rightarrow C$$. This result is in contrast to the conjectured answer ‘yes’ in the open problem: $$\inf\{ h_{\text{top}} (T)\mid h_{\text{top}} (T)>0\} >0$$? (where $$h_{\text{top}} (T)$$ is the topological entropy of $$T$$).

### MSC:

 37C35 Orbit growth in dynamical systems 22D40 Ergodic theory on groups 37B40 Topological entropy
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